Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings

0.35 MB·English

by  Manuel de Leon

HAMILTONIAN DYNAMICS AND CONSTRAINED
VARIATIONAL CALCULUS: CONTINUOUS AND
DISCRETE SETTINGS
MANUEL DELEO´N, FERNANDOJIME´NEZ, ANDDAVIDMART´IN DEDIEGO
2
1
Abstract. The aim of this paper is to study therelationship between
0
Hamiltonian dynamics and constrained variational calculus. We de-
2
scribe both using the notion of Lagrangian submanifolds of convenient

n symplectic manifolds and using the so-called Tulczyjew’s triples. The
a
results are also extended to thecase of discrete dynamicsand nonholo-
J
nomic mechanics. Interesting applications to geometrical integration of
1
Hamiltonian systems are obtained.

]
h 1. Introduction
p
- One of the main notions in symplectic geometry is the concept of La-
h
t grangian submanifolds. Thisconcept arises in several anddifferent interpre-
a
tations of physical, engineeringandgeometric phenomena. Inthispaper,we
m
willfocus ourattention intheir applications toLagrangian andHamiltonian
[
dynamics of constrained systems.

2 For instance, the theory of Lagrangian submanifolds gives a geometric
v
and intrinsic description of Lagrangian and Hamiltonian dynamics [34, 35].
0
7 Moreover, it allows us to relate both formalisms using as a main tool the
5 so-called Tulczyjew’s triple
5
8. T∗TM oo αM TT∗M βM // T∗T∗M.
0
1 Recall that, in the above expression, αM is the Tulczyjew’s canonical sym-
1 plectomorphism from TT∗M (with its canonical symplectic structure
: ∗
v d ω ) to T TM (equipped now with its canonical symplectic structure
T M
i ω ); and β is the canonical symplectomorphism defined by the sym-
X TM M
∗
plectic structure ω on T M. The Lagrangian dynamics is “generated”
r M
a by the Lagrangian submanifold dL(TM) of T∗TM where L : TM → R is
the Lagrangian function, while the Hamiltonian formalism is generated by
the Lagrangian submanifold dH(T∗M) of T∗T∗M where H : T∗M → R
is the corresponding Hamiltonian energy. The dynamics and the relation-
ship between both formalisms are based on the central part of the Tulczy-
jew’s triple, TT∗M, where the Lagrangian submanifolds α (dL(TM)) and
M
β−1(dH(T∗M)) live. Of course, any submanifold of a tangent bundle auto-
M
matically determines a system of implicit differential equations; in this case,
we can apply the integrability constraint algorithm, described in § 2.2 (see
[9, 32] for more details), to find the integrable part of the dynamics defined
by this Lagrangian submanifold.
This model is also valid for constrained variational calculus, determined
by a function L : C → R where C is a submanifold of TM, with inclusion
1
2 M.DELEO´N,F.JIME´NEZ,ANDD.MART´INDEDIEGO
i : C → TM. In this case, we can also construct a new Lagrangian
C
∗
submanifold (see § 4) Σ of T TM as:
L
∗ ∗
Σ = {µ ∈ T TM| i µ = dL}.
L C
Thus, we can obtain via α a new Lagrangian submanifold of the tangent
M
∗
bundle TT M which completely determines the equations of motion of the
constrained dynamics (see [16]), which are, in aregular case, of Hamiltonian
type. In this paper, we will also prove that given an arbitrary Hamiltonian
system we can construct a (possibly) constrained Lagrangian system that
generates the original one.
It is necessary to stress that the equations derived are purely variational
and, consequently, different from the nonholonomic equations obtained by
applyingtheLagrange-D’Alembert’sorChetaev’sprinciples(see[5,8,23]for
more details). It is well-known that the nonholonomic equations give the
right physical dynamics of a constrained mechanical systems [26], mainly
related to the rolling motion. Geometrically, nonholonomic constraints are
globally described by a submanifold C of the velocity phase space TM.
If C is a vector subbundle of TM, we are dealing with linear constraints
and, when C is an affine subbundle, we are in the case of affine constraints.
Lagrange-D’Alembert’s or Chetaev’s principles allow us to determine the
set of possible values of the constraint forces only from the set of admissible
kinematic states, that is, from the constraint submanifold C determined
by the vanishing of the nonholonomic constraints. An interesting study of
nonholonomic systems as implicit differential equations is presented in [19].
On the other hand, constrained variational calculus is mainly related to
mathematical and engineering applications, specially in control theory. In
thispaper,wewillshowthecloserelationship betweenclassical Hamiltonian
dynamics and constrained variational calculus. In fact, both are equivalent
under some regularity conditions. Nevertheless, nonholonomic mechanics is
∗
described by a general submanifold of TT M which is not Lagrangian; this
fact implies the non-preservation properties of the nonholonomic flow.
Moreover, we will study the discrete formalism, which will be also inter-
pretedin thesameway as Lagrangian submanifoldsof thecartesian product
∗
oftwocopiesofT M,equippedwithasuitablesymplecticstructure. Weare
interested in finding a geometrical answer to two, in principle, alternative
ways to derive geometric integrators for constrained Lagrangian problems:
a pure discrete variational procedure or a symplectic numerical method for
the associated Hamiltonian system. We will show in which cases both pro-
cedures match. Additionally, we will derive a general algorithm to compare
the dynamics in variational constrained and nonholonomic cases, both in
continuous and discrete cases.
2. Geometric preliminaries
2.1. Lagrangian submanifolds. In this section we will introduce some
particular constructions of Lagrangian submanifolds that are interesting for
our purposes (see [27, 36]).
HAMILTONIAN DYNAMICS AND CONSTRAINED VARIATIONAL CALCULUS 3
First, let us recall that given a finite-dimensional symplectic manifold
(P,ω) and a submanifold N, with canonical inclusion i : N ֒→ P, then N
N
is a Lagrangian submanifold if i∗ ω = 0 and dimN = 1dimP.
N 2
i) AninterestingclassofLagrangiansubmanifolds,whichwillbeuseful
in § 5, is the following. Let (P,ω) be a symplectic manifold and g :
P → P a diffeomorphism. Denote by Graph(g) the graph of g, that
isGraph(g) = {(x,g(x)), x ∈ P}⊂ P×P,andbypr : P×P → P,
i
i = {0,1}, the canonical projections. Then (P ×P,Ω), where Ω =
∗ ∗
pr ω−pr ω, is a symplectic manifold. Let i : Graph(g) ֒→ P ×P
1 0 g
be the inclusion map, then
∗ ∗ ∗
i Ω = (pr ) (g ω−ω).
g 0
∗
Thus, g is a symplectomorphism (that is, g ω = ω) if and only if
Graphg is a Lagrangian submanifold of P ×P.
∗
A distinguished symplectic manifold is the cotangent bundle T M of any
manifold M. If we choose local coordinates (qi), 1 ≤ i ≤ n, then T∗M
has induced coordinates (qi,p ). Denote by π : T∗M → M the canonical
i M
∗
projection defined by π (ǫ ) = q, where ǫ ∈ T M. Define the Liouville
M q q q
one-form or canonical one-form θ ∈ Λ1T∗M by
M
∗ ∗
h(θ ) , Xi = hǫ, Tπ (X)i, where X ∈ T T M , ǫ ∈ T M.
M ǫ M ǫ
In local coordinates we obtain θ = p dqi. The canonical two-form ω on
M i M
T∗M is the symplectic form ω = −dθ (that is ω = dqi∧dp ).
M M M i
ii) Now, we will introduce some special Lagrangian submanifolds of
∗
the symplectic manifold (T M,ω ). For instance, the image Σ =
M λ
λ(M) ⊂ T∗M of a closed one-form λ ∈ Λ1M is a Lagrangian sub-
∗ ∗
manifold of (T M,ω ), since λ ω = −dλ. We then obtain a sub-
M M
∗
manifold diffeomorphic to M and transverse to the fibers of T M.
When λ is exact, that is, λ = df, where f : M → R, we say that
f is a generating function of the Lagrangian submanifold Σ = Σ .
λ f
Locally, this is always the case.
A useful extension of the previous construction is the following result due
to W.M. Tulczyjew.
Theorem 2.1 ([34],[35]). Let M be a smooth manifold, τ : TM → M its
M
tangent bundle projection, N ⊂ M a submanifold, and f: N → R. Then
∗
Σ = p ∈ T M |π (p) ∈N and hp,vi = hdf,vi
f M
(cid:8) for all v ∈TN ⊂ TM such that τM(v) = πM(p)
∗
is a Lagrangian submanifold of T M. (cid:9)
Taking f as the zero function we obtain the following Lagrangian sub-
manifold
∗
Σ = p ∈ T M |hp, vi = 0, ∀v ∈ TN withτ (v) = π (p) ,
0 N M M
which is jus(cid:8)t the cono(cid:12)rmal bundle of N: (cid:9)
(cid:12)
∗ ∗
ν (N) = ξ ∈ T M ; ξ = 0 .
N T N
π(ξ)
n (cid:12) (cid:12) o
(cid:12) (cid:12)
4 M.DELEO´N,F.JIME´NEZ,ANDD.MART´INDEDIEGO
Given a symplectic manifold (P,ω), dimP = 2n it is well-known that its
tangent bundleTP is equipped with a symplectic structure denoted by d ω
T
(see [25]). If we take Darboux coordinates (qi,p ) on P, 1 ≤ i ≤ n, then
i
ω = dqi∧dp and, consequently, we have induced coordinates (qi,p ;q˙i,p˙ ),
i i i
(qi,p ;a ,bi)on TP andT∗P, respectively. Thus,d ω = dq˙i∧dp +dqi∧dp˙
i i T i i
and ω = dqi ∧da +dp ∧dbi. If we denote by ♭ : TP → T∗P the iso-
P i i ω
morphism defined by ω, that is ♭ (v) = i ω, then we have ♭ (qi,p ;q˙i,p˙ ) =
ω v ω i i
(qi,p ;−p˙ ,q˙i). GivenafunctionH : P → R,anditsassociated Hamiltonian
i i
vector field X , that is, i ω = dH, the image X (P) is a Lagrangian
H XH P H
submanifold of (TP,d ω ). Moreover, given a vector field X ∈ X(P), it
T P
is locally Hamiltonian if and only if its image X(P) is a Lagrangian sub-
∗
manifold of (TP,d ω). It is interesting to note that d ω = −♭ ω and
T T ω M
♭ (X (M)) = dH(M).
ω H
As it is briefly mentioned above, an important notion in the theory of
Lagrangian submanifolds is the concept of generating function. If we have
a Lagrangian submanifold N of an exact symplectic manifold (P,ω = dθ),
where θ ∈ Λ1P, then 0 = i∗ ω = i∗ dθ = d(i∗ θ). Consequently, applying
N N N
the Poincar´e’s lemma, there exists a function S : U → R defined on a
∗
open neihborhood U of N such that i θ = dS. We say that S is a (local)
N
generating function of the Lagrangian submanifold N.
2.2. Implicit differential equations. An implicit differential equation on
a general smooth manifold M is a submanifold E ⊂ TM. A solution of E
is any curve γ :I → M, I ⊂ R, such that the tangent curve (γ(t),γ˙(t)) ∈E
for all t ∈ I. The implicit differential equation will be said to be integrable
at a point if there exists a solution γ of E such that thetangent curve passes
through it. Furthermore, the implicit differential equation will be said to be
integrable if it is integrable at all points. Unfortunately, integrability does
notmean uniqueness. Theintegrable partofE isthesubsetof all integrable
points of E. The integrability problem consists in identifying such a subset.
Denoting the canonical projection τ : TM → M, a sufficient condition
M
for the integrability of E is
E ⊂ TM,
where C = τ (E), provided that the projection τ restricted is a submer-
M M
sion onto C.
2.2.1. Extracting the integrable part of E. A recursive algorithm was pre-
sented in [32] that allows to extract the integrable part of an implicit differ-
ential equation E. We shall define the subsets
E = E, C = C,
0 0
and recursively for every k ≥ 1,
Ek = Ek−1∩TCk−1, Ck = τM(Ek),
then, eventually the recursive construction will stabilize in the sense that
Ek = Ek+1 = ... = E∞, and Ck = Ck+1 = ... = C∞. It is clear by
construction that E∞ ⊂ TC∞. Then, provided that the adequate regularity
conditions are satisfied during the application of the algorithm, the implicit
differential equations E∞ willbeintegrable andit willsolve theintegrability
problem
HAMILTONIAN DYNAMICS AND CONSTRAINED VARIATIONAL CALCULUS 5
3. Tulczyjew’s triples
In this section we summarize a classical result due to W.M. Tulczyjew
∗ ∗
showing a natural identification of T TM and TT M, where M is any
smooth manifold, as symplectic manifolds. This construction plays a key
role in Lagrangian and Hamiltonian mechanics.
∗ ∗ ∗ ∗
Is easy to see that TT M, T TM and T T M are naturally double vec-
∗
tor bundles (see [12], [33]) over T M and TM. In [34] and [35], Tulczyjew
∗ ∗
establishedtwoidentifications,thefirstonebetweenTT M andT TM (use-
∗
ful to describe Lagrangian mechanics) and the second one between TT M
∗ ∗
and T T M (useful to describe Hamiltonian mechanics). The Tulczyjew
∗ ∗
map α is an isomorphism between TT M and T TM. Beside, it is also
M
a symplectomorphism between these double vector bundles as symplectic
∗
manifolds, i.e. (TT M , d ω ), where d ω is the tangent lift of ω , and
T M T M M
∗
(T TM,ω ). In the following diagram we show the different relationships
TM
among these bundles.
TT∗M αM // T∗TM
✾ ✿
☎ ✾ ✆ ✿
☎ ✾ ✆ ✿
☎ ✾ ✆ ✿
☎ ✾ ✆ ✿
☎ ✾ ✆ ✿
τT∗M ☎☎☎ ✾✾✾TπM πTM ✆✆✆ ✿✿✿T∗τM
☎ ✾ ✆ ✿
☎ ✾ ✆ ✿
☎ ✾ ✆ ✿
☎ ✾ ✆ ✿
☎ ✾ ✆ ✿
☎ ✾ ✆ ✿
(cid:1)(cid:1)☎ (cid:28)(cid:28) (cid:2)(cid:2)✆ (cid:29)(cid:29)
∗ ∗
T M TM T M
▲▲▲▲▲▲▲▲▲π▲M▲▲▲▲▲▲▲▲▲▲▲▲τ▲M&& (cid:15)(cid:15) yyrrrrrrrrrrrrπrMrrrrrrrrrr
M
∗
The definition of T τ is given in the following remark.
M
Remark 3.1. Given a tangent bundle τ : TN → N, for each y ∈ T N we
N x
can define
V = ker{T τ :T TN → T N}, τ (y) = x.
y y N y x N
Summing over all y we obtain a vector bundle V of rank n over TN. Any
element u ∈ T N determines a vertical vector at any point y in the fibre
x
over x, called its vertical lift to y, denoted by uV(y). It is the tangent vector
at t = 0 to the curve y +tu. If X is a vector field on N, we may define
its vertical lift as XV(y) = (X(τ (y)))V. Locally, if X = Xi ∂ in a local
N ∂xi
neighborhood U with local coordinates xi, then XV is locally given by
∂
XV = Xi ,
∂vi
with respect to induced coordinates (xi,vi) on TU.
Now, we define T∗τ : T∗TM → T∗M by hT∗τ (α ),wi = hα ,wVi;
M M u u u
u,w ∈ T M, α ∈ T∗TM and wV ∈ T TM.
q u u u u
In the following, we recall the construction of the symplectomorphism
α . To do this, it is necessary to introduce the canonical flip ([12]) on
M
6 M.DELEO´N,F.JIME´NEZ,ANDD.MART´INDEDIEGO
TTM:
TTM // TTM
κM
τTM TτM
(cid:15)(cid:15) (cid:15)(cid:15)
TM // TM,
Id
as follows:
d d d d
κ χ(s,t) = χ˜(s,t),
M
ds s=0dt t=0 ds s=0dt t=0
(cid:18) (cid:12) (cid:12) (cid:19) (cid:12) (cid:12)
where χ : R2 → M(cid:12)and χ˜(cid:12): R2 → M are re(cid:12)lated b(cid:12)y χ˜(s,t) = χ(t,s). If
(cid:12) (cid:12) (cid:12) (cid:12)
qi are the local coordinates for M, qi,vi for TM and qi,vi,q˙i,v˙i for
TTM, then the canonical involution can be defined as κ qi,vi,q˙i,v˙i =
(cid:0) (cid:1) (cid:0) (cid:1) M(cid:0) (cid:1)
qi,q˙i,vi,v˙i .
(cid:0) (cid:1)
In order to describe α is also necessary to define a tangent pairing.
M
(cid:0) (cid:1)
Given twomanifoldsM andN, andapairingbetween themh·,·i :M×N →
R, the tangent pairing h·,·iT :TM ×TN → R is determined by
d d d
h γ(t), δ(t)iT = hγ(t),δ(t)i
dt t=0 dt t=0 dt t=0
where γ : R → M(cid:12)(cid:12) and δ : R→(cid:12)(cid:12) N. (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12)
Finally, we can define α as hα (z),wi = hz,κ (w)iT, where z ∈
M M M
∗
TT M and w ∈ TTM. In local coordinates:
α qi,p ,q˙i,p˙ = qi,q˙i,p˙ ,p ;
M i i i i
now qi,p are the local(cid:0)coordinates(cid:1)for(cid:0)T∗M and (cid:1)qi,p ,q˙i,p˙ for TT∗M.
i i i
∗ ∗ ∗
ThethirddoublevectorbundleisT T M. Theisomorphismβ : TT M →
M
∗ ∗(cid:0) (cid:1) (cid:0) (cid:1)
T T M is just given by β = ♭ (see previous subsection).
M ∗ωM ∗ ∗ ∗
Considering the bundles TT M, T TM and T T M, as well as the sym-
plectomorphisms α and β , we finally obtain the Tulczyjew’s triple
M M
T∗T∗M oo βM TT∗M αM // T∗TM .
4. Continuous Lagrangian and Hamiltonian mechanics
In the introduction we have shown how the Tulcyjew’s triple is used to
describe geometrically Lagrangian and Hamiltonian mechanics and its re-
lationship. In this section we will see that it is also possible to adapt this
geometric formalism when we introduce constraints into the picture. As we
have mentioned along the introduction, there are (at least) two methods
that one might use to derive the equations of motion of systems subjected
to constraints. We will call them nonholonomic mechanics and constrained
variational calculus. The classical method to derive equations of motion for
constrained mechanical systems is the nonholonomic mechanics. The equa-
tions derived from the nonholonomic methods are not of variational nature,
but they describe the correct dynamics of a constrained mechanical system.
Inordertoobtainthenonholonomicequations,ifwehavelinearoraffinecon-
straints, is necessary to apply the Lagrange-D’Alembert’s principle. When
dealing with nonlinear constraints, one should employ the more controver-
sial Chetaev’s rule (see [5, 23] for further details). Since the geometrical
HAMILTONIAN DYNAMICS AND CONSTRAINED VARIATIONAL CALCULUS 7
implementation of the Chetaev’s rule is practically equal to the process in
the linear case, we shall use it from a pure mathematical perspective.
On the other hand, the equations of motion of constrained variational
problems are derivable by using variational techniques (always in the con-
strained case). These last equations are also known in the literature as
vakonomic equations. The terminology vakonomic (“mechanics of varia-
tional axiomatic kind”) was coined by V.V. Kozlov ([3],[22]). The main
applications of the constrained variational calculus appear in problems of
mathematical nature (like subriemannian geometry) and in optimal control
theory.
4.1. Nonholonomic mechanics. A nonholonomic system on a manifold
M consists of a pair (L,C), where L : TM → R is the Lagrangian of the
mechanical system and C is a submanifold of TM with canonical inclusion
i :C ֒→ TM. In the following, we will assume, for sake of symplicity, that
C
τ (C) = M. Since the motion of the system is forced to take place on the
M
submanifold C, this requires the introduction of some reaction or constraint
forces into the system. If φα(qi,q˙i) = 0, 1 ≤ α ≤ n, determine locally the
submanifold C, then Chetaev’s rule implies that the constrained equations
of the system are:
d ∂L ∂L ∂φα
− = λ ,
dt ∂q˙i ∂qi α ∂q˙i
(cid:18) (cid:19)
(1)
φα(qi,q˙i) = 0, 1 ≤ α ≤ n.
Next, we will describe geometrically the nonholonomic equations. First, we
need to introduce the vertical endomorphism S which is a (1,1)-tensor field
on TM defined by
S : TTM −→ TTM
d
W 7−→ (v +tTτ (W )).
vx dt t=0 x M vx
(cid:12)
Its local expression is S = ∂ ⊗dqi.(cid:12)
∂q˙i (cid:12)
If we accept Chetaev-type forces, then we define
F = S∗(TC)0.
Observe that the vector subbundleF will be generated by the 1-forms µα =
∂φα dqi because S∗(dφα) = µα.
∂q˙i
∗
Now, define the affine subbundle of T TM given by
C
Σnoh = (dL)◦i +F,
C
that is,
Σnoh = {(qi,q˙i,µ ,µ˜ ) ∈ T∗TM | (2)
i i
∂L ∂φα
µ = +λ ,
i ∂qi α ∂q˙i
∂L
µ˜ = ,
i ∂q˙i
φα(q,q˙)= 0, 1 ≤ α ≤ n}.
8 M.DELEO´N,F.JIME´NEZ,ANDD.MART´INDEDIEGO
Therefore, applying the Tulczyjew’s isomorphism α we obtain the affine
M
subbundle
α−1 Σnoh = {(qi,p ,q˙i,p˙ )∈ TT∗M | (3)
M i i
(cid:16) (cid:17) ∂L
p = ,
i ∂q˙i
∂L ∂φα
p˙ = +λ ,
i ∂qi α ∂q˙i
φα(q,q˙)= 0, 1≤ α ≤ n}.
Define now the nonholonomic Legendre transformation FLnoh : C → T∗M
by
FLnoh = πT∗M ◦α−M1◦dL◦iC .
The solutions for the dynamics given by α−1 Σnoh are curves σ : I ⊂ R →
M
M suchthat dσ(I) ⊂ C andtheinducedcurveγ : R → T∗M,γ = FLnoh(dσ)
dt (cid:0) (cid:1) dt
verifies that dγ(I) ⊂ α−1 Σnoh . Locally, σ must satisfy the system of
dt M
equations (1).
(cid:0) (cid:1)
An interesting use of Tulczyjew’s triple in order to define Lagrangian
submanifoldsandgeneralized Legendre transformations within thenonholo-
nomic framework can be found in [31].
4.2. Variational constrained equations. Now, we study the same prob-
lem but now using purely variational techniques. As above, let consider a
regular Lagrangian L : TM → R, and a set of nonholonomic constraints
φα(qi,q˙i), 1 ≤ α ≤ n, determining a 2m − n dimensional submanifold
C ⊂ TM. Now we take the extended Lagrangian L = L + λ φα which
α
includes the Lagrange multipliers λ as new extra variables. The equations
α
of motion for the constrained variational problem are the Euler-Lagrange
equations for L, that is:
d ∂L ∂L ∂φα d ∂φα ∂φα
− = −λ˙ −λ − ,
dt ∂q˙i ∂qi α ∂q˙i α dt ∂q˙i ∂qi
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:21)
(4)
φα(qi,q˙i) = 0, 1≤ α ≤ n.
From a geometrical point of view, these type of variationally constrained
problems are determined by a pair (C,L) where C is a submanifold of
TM, with inclusion i : C ֒→ TM, and L : C → R a Lagrangian func-
C
tion. Using Theorem 2.1 we deduce that Σ is a Lagrangian submanifold
L
∗
of (T TM,ω ) (see [16]). Now using the Tulczyjew’s symplectomorphism
TM
α , we induce a new Lagrangian submanifold α−1(Σ ) of (TT∗M,d ω ),
M M L T M
whichcompletelydeterminestheconstrainedvariationaldynamics. Ofcourse,
the case of unconstrained Lagrangian mechanics is generated taking the
whole space TM instead of C and an a Lagrangian function over the tan-
gent bundle L :TM → R.
Next we shall prove that, indeed, this procedure gives the correct equa-
tions for the constrained variationalpp dynamics. Take an arbitrary exten-
sion L :TM → R of L : C → R, that is, L◦i = L. As above, assume also
C
HAMILTONIAN DYNAMICS AND CONSTRAINED VARIATIONAL CALCULUS 9
that we have fixed local constraints such that locally determines C by their
vanishing, i.e: φα(q,q˙)= 0, 1 ≤ α ≤ n where n= 2dim M −dim C.
Locally
Σ = {(qi,q˙i,µ ,µ˜ )∈ T∗TM | (5)
L i i
∂L ∂φα
µ = +λ ,
i ∂qi α ∂qi
∂L ∂φα
µ˜ = +λ ,
i ∂q˙i α ∂q˙i
φα(q,q˙)= 0, 1≤ α ≤ n}.
Observe that locally the conormal bundle ν∗(C)= span {dφα,1 ≤ α≤ n}.
Therefore,
α−1(Σ ) = {(qi,p ,q˙i,p˙ )∈ TT∗M | (6)
M L i i
∂L ∂φα
p = +λ ,
i ∂q˙i α ∂q˙i
∂L ∂φα
p˙ = +λ ,
i ∂qi α ∂qi
φα(q,q˙) = 0, 1 ≤ α≤ n} .
The solutions for the dynamics given by α−1(Σ ) ⊂ TT∗M are curves
M L
γ : I ⊂ R → T∗M such that dγ : I ⊂ R → TT∗M verifies that dγ(I) ⊂
dt dt
α−1(Σ ). Locally, if γ(t) = (qi(t),p (t)) then it must verify the following
M L i
set of differential equations:
d ∂L ∂φα ∂L ∂φα
+λ − −λ = 0,
dt ∂q˙i α ∂q˙i ∂qi α ∂qi
(cid:18) (cid:19)
φα(qi,q˙i) = 0,
which clearly coincide with equations (4).
Now,weconsideradaptedcoordinates(qi,q˙a)tothesubmanifoldC (recall
that τ (C) = M is now a fibration C → M), 1 ≤ i ≤ dimM and 1 ≤ a ≤
M
dimM −n, such that
i (qi,q˙a) = qi,q˙a,Ψα(qi,q˙a) .
C
(cid:0) (cid:1)
This means that φα(qi,q˙i) = q˙α−Ψα(qi,q˙a)= 0. Therefore, we have
Σ = {(qi,q˙i,µ ,µ˜ )| (7)
L i i
∂L ∂Ψα
µ = −µ˜ ,
i ∂qi α ∂qi
∂L ∂Ψα
µ˜ = −µ˜ ,
a ∂q˙a α ∂q˙a
q˙α = Ψα(qi,q˙a), 1≤ α ≤ n}.
Observe that (qi,q˙a,µ˜ ) determines a local system of coordinates for Σ .
α L
10 M.DELEO´N,F.JIME´NEZ,ANDD.MART´INDEDIEGO
Then,
α−1(Σ ) = {(qi,p ,q˙i,p˙ )∈ TT∗M | (8)
M L i i
∂L ∂Ψα
p = −p ,
a ∂q˙a α ∂q˙a
∂L ∂Ψα
p˙ = −p ,
i ∂qi α ∂qi
q˙α = Ψα(qi,q˙a), 1 ≤ α≤ n} .
Consequently, the solutions must verify the following system of differential
equations (see [9]):
d ∂L ∂Ψα ∂L ∂Ψα
−p = −p
dt ∂q˙a α ∂q˙a ∂qa α ∂qa
(cid:18) (cid:19)
∂L ∂Ψα
p˙ = −p ,
β ∂qβ α ∂qβ
q˙α = Ψα(qi,q˙a), 1 ≤ α≤ n .
4.2.1. The constrained Legendre transformation.
Definition 4.1. We define the constrained Legendre transformation FL :
ΣL −→ T∗M as the mapping FL= τT∗M ◦(α−M1)|ΣL.
We will say that the constrained system (L,C) is regular if FL is a local
diffeomorphism and hyperregular if FL is a global diffeomorphism.
Observethatlocally,ifasaboveweconsidertheconstraintsq˙α =Ψα(qi,q˙a)
determining C, then
∂L ∂Ψα
FL(qi,q˙a,µ˜ ) = (qi,p = −µ˜ ,p = µ˜ ).
α a ∂q˙a α ∂q˙a α α
The constrained system (L,C) is regular if and only if ∂2L −µ˜ ∂2Ψα
∂q˙a∂q˙b α∂q˙a∂q˙b
is a nondegenerate matrix. (cid:16) (cid:17)
Next, define the energy function E : Σ → R by
L L
E (α ) = hα ,uVi−L(u), α ∈Σ ,u ∈ C ≡i (C)
L u u u u L C
Locally, we have
∂L ∂Ψα(qi,q˙a)
E (qi,q˙a,µ˜ ) = q˙a −µ˜ q˙a+µ˜ Ψα(qi,q˙a)−L(qi,q˙a).
L α ∂q˙a α ∂q˙a α
Remark 4.2. The constrained Legendre transformation allows us to de-
velop a Lagrangian formalism on Σ . Indeed, we can define the 2-form
L
ω = (FL)∗ω on Σ and it is easy to show that the equations of motion
L M L
of the constrained system are now intrinsically rewritten as
i ω = dE .
X L L
In consequence, we could develop an intrinsic formalism on the Lagrangian
side, that is a Klein formalism ([12], [17], [21], [25]) for constrained systems
without using (at least initially) Lagrangian multipliers. Moreover, notice
thattheconstrainedsystemisregularifandonlyifω isasymplectic2-form
L
on Σ
L